# Chapter 3: An Introduction to Probability

**Advanced**Created by: CK-12

## Introduction

The concept of probability plays an important role in our daily lives. Assume you have an opportunity to invest some money in a software company. Suppose you know that the company’s records indicate that in the past five years, its profits have been consistently decreasing. Would you still invest your money in it? Do you think the chances are good for the company in the future?

Here is another illustration. Suppose that you are playing a game that involves tossing a single die. Assume that you have already tossed it 10 times, and every time the outcome was the same, a 2. What is your prediction of the eleventh toss? Would you be willing to bet $100 that you will not get a 2 on the next toss? Do you think the die is loaded?

Notice that the decision concerning a successful investment in the software company and the decision of whether or not to bet $100 on the next outcome of the die are both based on probabilities of certain sample results. Namely, the software company’s profits have been declining for the past five years, and the outcome of rolling a 2 ten times in a row seems strange. From these sample results, we might conclude that we are not going to invest our money in the software company or bet on this die. In this chapter, you will learn mathematical ideas and tools that can help you understand such situations.

- 3.1.
## Sample Spaces and Events

- 3.2.
## Operations with Sets

- 3.3.
## Complement Rule for Probability

- 3.4.
## Conditional Probability

- 3.5.
## Additive and Multiplicative Rules for Probability

- 3.6.
## Basic Counting Rules

- 3.7.
## Permutations and Combinations Compared

### Chapter Summary

## Summary

This chapter begins by introducing students to events, sample spaces and probabilities. It then continues by explaining more complex sample spaces and probabilities including complements and conditional probabilities, as well as independent and mutually exclusive events. Additionally, it introduces students to counting possibilities using the Multiplication Rule for Counting, as well as how to count the number of permutations and combinations.

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